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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a350 at Knotilus!

Link Presentations

[edit Notes on L11a350's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,7,21,8 X6,21,7,22 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17
Gauss code {1, -10, 2, -7, 8, -6, 5, -2, 9, -11}, {10, -1, 3, -8, 7, -9, 11, -3, 4, -5, 6, -4}
A Braid Representative
A Morse Link Presentation L11a350 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+4 t(2) t(1)^3-2 t(1)^3-3 t(2)^3 t(1)^2+8 t(2)^2 t(1)^2-8 t(2) t(1)^2+4 t(1)^2+4 t(2)^3 t(1)-8 t(2)^2 t(1)+8 t(2) t(1)-3 t(1)-2 t(2)^3+4 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+9 q^{7/2}-15 q^{5/2}+19 q^{3/2}-21 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-5 a^3 z+6 a z-6 z a^{-1} +2 z a^{-3} +a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-4 a^3 z^9-12 a z^9-8 z^9 a^{-1} -3 a^4 z^8-6 a^2 z^8-14 z^8 a^{-2} -17 z^8-a^5 z^7+9 a^3 z^7+25 a z^7+z^7 a^{-1} -14 z^7 a^{-3} +10 a^4 z^6+32 a^2 z^6+22 z^6 a^{-2} -9 z^6 a^{-4} +53 z^6+4 a^5 z^5+a^3 z^5-a z^5+26 z^5 a^{-1} +20 z^5 a^{-3} -4 z^5 a^{-5} -10 a^4 z^4-32 a^2 z^4-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -40 z^4-6 a^5 z^3-11 a^3 z^3-14 a z^3-21 z^3 a^{-1} -11 z^3 a^{-3} +z^3 a^{-5} +2 a^4 z^2+9 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +11 z^2+4 a^5 z+5 a^3 z+4 a z+5 z a^{-1} +2 z a^{-3} +a^4-a^5 z^{-1} -a^3 z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
j \
12           1-1
10          3 3
8         61 -5
6        93  6
4       106   -4
2      119    2
0     1111     0
-2    710      -3
-4   511       6
-6  37        -4
-8 16         5
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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